rowvec dF2du(unsigned row, irowvec causes, const DataPairs &data, const gmat &sigma, vec u){
// Attaining variance covariance matrix etc. (conditional on u)
vmat cond_sig = sigma(causes);
vec cond_mean = cond_sig.proj*u;
rowvec alp = data.alpha_get(row, causes);
rowvec gam = data.gamma_get(row, causes);
colvec alpgam = alp.t() - gam.t();
double cdf = pn(alpgam, cond_mean, cond_sig.vcov);
vec ll = sqrt(diagvec(cond_sig.vcov));
mat Lambda = diagmat(ll);
mat iLambda = diagmat(1/ll);
mat R = iLambda*cond_sig.vcov*iLambda;
mat LR = Lambda*R;
double r = R(0,1);
vec ytilde = iLambda*(alpgam - cond_mean);
vecmat D = Dbvn(ytilde(0),ytilde(1),r);
mat M = -LR*D.V;
vec dcdfdu = trans(cond_sig.proj)*cond_sig.inv*M;
rowvec dF2du_1 = data.dpiduMarg_get(row, causes(0), 0)*data.piMarg_get(row, causes(1), 1)*cdf ;
rowvec dF2du_2 = data.dpiduMarg_get(row, causes(1), 1)*data.piMarg_get(row, causes(0), 0)*cdf;
vec dF2du_3 = data.piMarg_get(row, causes(0), 0)*data.piMarg_get(row, causes(1), 1)*dcdfdu;
rowvec dF2du = dF2du_1 + dF2du_2 + dF2du_3.t();
return(dF2du);
};
void SetImage::pave(const IntervalVector& x, double epsilon) {
Linside.clear();
Lboundary.clear();
PdcImageSubset p_fin(f,x,p_in);
assert(x.size()==n);
stack<IntervalVector> Ldomain;
IntervalVector xtilde(n);
IntervalVector ytilde(n);
LargestFirst lf(epsilon);
Ldomain.push(x);
while (! Ldomain.empty()) {
xtilde = Ldomain.top();
Ldomain.pop();
c_out.contract(xtilde);
if (xtilde.is_empty()) {
continue;
}
// use natural extension
ytilde=f.eval_vector(xtilde);
// improve with centered form
ytilde&=f.eval_vector(xtilde.mid())+f.jacobian(xtilde)*(xtilde-xtilde.mid());
if (p_in.test(xtilde)==YES && p_fin.test(cart_prod(xtilde,ytilde))==YES)
Linside.push_back(ytilde);
else if (xtilde.max_diam()<=epsilon)
Lboundary.push_back(ytilde);
else {
pair<IntervalVector,IntervalVector> boxes=lf.bisect(xtilde);
Ldomain.push(boxes.first);
Ldomain.push(boxes.second);
}
}
}
